In YOLO, what exactly do the values associated with each anchor box represent?
Not an pro but I think I know some answers to your questions. 


  If we train our classifier, wouldn't the prediction boxes be close to the ground truth labels as training progresses


I think that's what YOLO v1 did. According to Andrew NG's video the bounding boxes are introduced to solve multiple objects inside the same grid cell. And according to this post anchor boxes assignment ensures that an anchor box predicts ground truth for an object centered at its own grid center, and not a grid cell far away (like YOLO may)


  what are those numbers representing anchor boxes representing? 


They are just width and height (shape). In YOLO v2 it is used to compute IOU assuming all boxes are placed at the same location (ignoring the location), you could think of it just tries to match the shape. And it uses (1-IOU) as the distance when applying the K-means clustering.